Simultaneity as an Invariant Equivalence Relation

Abstract

This paper deals with the concept of simultaneity in classical and relativistic physics as construed in terms of group-invariant equivalence relations. A full examination of Newton, Galilei and Poincaré invariant equivalence relations in ℝ⁴ is presented, which provides alternative proofs, additions and occasionally corrections of results in the literature, including Malament’s theorem and some of its variants. It is argued that the interpretation of simultaneity as an invariant equivalence relation, although interesting for its own sake, does not cut in the debate concerning the conventionality of simultaneity in special relativity.

Appendix

We prove Lemma 3.4, that is:

Lemma 3.4

All proper subgroups H of ℝ are totally disconnected; they are either of the form ℤa, with a∈ℝ⁺, or everywhere dense.

Proof

It is enough to argue by restricting ourselves to the positive elements of ℝ. If H is nonzero, then H∩ℝ⁺ is nonempty; let a be its infimum. First suppose that a is nonzero. If a does not belong to H, then for positive ϵ<a there exist h ₁,h ₂∈H such that a<h ₁<h ₂<a+ϵ, whence h ₂−h ₁<ϵ<a is a positive element of H less than a, which is absurd. It follows that a is the positive minimum of H, so for every other b∈H∩ℝ⁺ it must be

$$b = ma +r, \quad \mbox{where}\ m\in\mathbb{Z}^+, 0\leq r <a. $$

It follows that b−ma is an element of $H\cap\mathbb{R}^{+}_{0}$ which is smaller than a, so it must be zero, and therefore b∈ℤa, that is H=ℤa.

Suppose now a=0; then there is a strictly decreasing sequence (h _n ) of elements in H which converges to zero. Let x be any positive real number. For every positive ϵ, no matter how small, there is n ₀ such that $h_{n_{0}} < \epsilon$. Thus by dividing x by $h_{n_{0}}$ we get, for a suitable n∈ℤ⁺ and r≥0 strictly smaller than $h_{n_{0}}$ : $x = nh_{n_{0}} + r$, and therefore $x - nh_{n_{0}} =r<\epsilon$, where of course $nh_{n_{0}}$ belongs to H. Thus in every neighborhood of every (positive) real number there lies some element of H, which means that H is everywhere dense in ℝ.

As to the topological structure, it is clear that every subgroup of the form ℤa is totally disconnected. Suppose now that H is everywhere dense; if the connected component K of 0 in H is not a singleton, then it must contain an interval [0,ϵ[ for some ϵ>0, but then it must also contain ]−ϵ,ϵ[ and therefore

$$K\supseteq\bigcup_{n\in\mathbb{N}} n]{-}\epsilon,\epsilon[ = \bigcup_{n\in\mathbb{N}} ]{-}n\epsilon, n \epsilon[ = \mathbb{R}, $$

which contradicts the hypothesis that H is proper. □